Computing Positive Semidefinite Minimum Rank for Small Graphs
نویسندگان
چکیده
Abstract. The positive semidefinite minimum rank of a simple graph G is defined to be the smallest possible rank over all 1 positive semidefinite real symmetric matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero 2 otherwise. The computation of this parameter directly is difficult. However, there are a number of known bounding parameters 3 and techniques which can be calculated and performed on a computer. We programmed an implementation of these bounds 4 and techniques in the open-source mathematical software Sage. The program, in conjunction with the orthogonal representation 5 method, establishes the positive semidefinite minimum rank for all graphs of order 7 or less. 6
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